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A(1,-3)

B(1,-1)
C(4,-3)

E(-2,5)
H(-4,2)
E(-2,5)

△ABC is congruent to △GHE. Describe a sequence of transformations from △ABC to △GHE that proves their congruence in a few shorts sentences

1 answer

To show that triangles \( \triangle ABC \) and \( \triangle GHE \) are congruent, we can apply the following sequence of transformations:

  1. Reflection: Reflect \( \triangle ABC \) over the y-axis. This transformation will change the coordinates of points \( A(1, -3) \) and \( B(1, -1) \) to \( A'(-1, -3) \) and \( B'(-1, -1) \), while point \( C(4, -3) \) becomes \( C'(-4, -3) \).

  2. Translation: Translate \( \triangle A'B'C' \) by moving it to the left by 1 unit and up by 5 units. After this translation:

    • \( A'(-1, -3) \) becomes \( A''(-2, 2) \)
    • \( B'(-1, -1) \) becomes \( B''(-2, 4) \)
    • \( C'(-4, -3) \) becomes \( C''(-5, 2) \)

  3. Final Adjustments: Rescale or rotate if necessary to match the coordinates exactly with points \( G, H, \) and \( E \).

After performing these transformations, \( \triangle ABC \) can be shown to be congruent to \( \triangle GHE \) based on the characteristics of congruent triangles, confirming that they have the same size and shape.